A single-determinant restricted open-shell Hartree-Fock (ROHF) wavefunction
describing a high-spin open-shell system will be an eigenfunction
of
(i.e., a CSF). This is easy to verify by direct
application of the
operator, which is

=

=

(46)

where

=

(47)

=

(48)

=

(49)

and

(50)

The high-spin ROHF wavefunction can be written as

(51)

where t and u will denote open-shells. Applying
,
this becomes

(52)

This is easy to evaluate:

=

(53)

=

(54)

The raising operator
yields zero when acting on
:
raising operators applied to
electrons
always yield zero, and raising operators applied to the electrons yield
spin orbitals which are already occupied (so
the determinant vanishes by the Pauli principle). Hence the final
result is

=

=

(55)

where
.

Now it is worthwhile to consider how to form CSFs for the single
excitations. Determinants which promote electrons from the
singly-occupied space to the virtual space, as well as determinants
which promote
electrons in the doubly-occupied orbitals to the
singly-occupied orbitals, are already spin-adapted (the proof is
completely analogous to that above for the ROHF reference
determinant). The only other relevant single excitations are those
which move an electron from a doubly-occupied orbital to a virtual
orbital. These determinants are not spin-adapted, as we will
proceed to demonstrate. Consider the action of
on the
determinant
:

(56)

The result of
is easily determined to be

(57)

Now all that remains is the raising and lowering operators. These are
somewhat more involved and require that attention be paid to the
sign.

(58)

By arguments similar to those presented above, all raising operators
yield zero except for
.
Using the
anticommutation relations for creation and annihilation operators,

=

=

=

=

(59)

The
operator can now affect electrons in any of the
following orbitals: a, i, and any of the open-shell orbitals.

=

=

(60)

Thus overall,

(61)

The analogous equation for
is

(62)

Clearly a spin eigenfunction can be constructed as

(63)

Then the operation of
is

(64)

Now that the relevant CSFs have been obtained, they can be used to
define the ROHF convergence criteria: the final ROHF wavefunction will
not mix with any of the singly substituted CSFs. Thus

(65)

which implies the following conditions on the Fock matrix elements

Fta

=

0

(66)

=

0

(67)

Fia

=

(68)

Using these results, we can write down expressions for the vectors. Since determinants
must enter with the same
coefficients as
,
once again. Furthermore, since the ROHF reference cannot mix
with any other singly excited configurations, the c0 contribution
to
and
may be
safely ignored. We will therefore consider four cases:
,
,
,
and
,
where once again t,u represent
singly occupied orbitals.

The equation for
is readily seen to be

=

+

(69)

Separating the Fock operator terms from the two-electron integrals,
and integrating out spin, this yields

=

+

(70)

where we have used the relation
.
The analogous equation for
spins is

=

+

(71)

If the equality
is to be
maintained, we must have
.
It is then computationally
convenient to form these quantities as

(72)

Thus

=

+

+

(73)

Now it is clear that the two-electron integrals can be treated all
together. To begin condensing the notation once again, let us define
the following quantities:

=

(74)

=

(75)

=

(76)

=

(77)

=

(78)

where
the same as that defined in eq. (17)
of Maurice and Head-Gordon [2].
Then
can be evaluated as

(79)

where

=

(80)

=

(81)

Next consider the term
:

=

=

+

=

(82)

It is computationally more efficient to evaluate
at the
same time as
.
The value of
computed in this
manner must of course be corrected for the difference in the formulas
for
and
,
but this correction scales as only
(see Maurice and Head-Gordon [2]). The
two-electron part is thus computed by